Let [.] denote the greatest integer function. Assertion (A) : $\lim _{x \rightarrow \infty} \frac{[x]}{x}=1$…

Let [.] denote the greatest integer function. Assertion (A) : $\lim _{x \rightarrow \infty} \frac{[x]}{x}=1$ Reason $(R)$ : $f(x)=x-1, g(x)=[x], h(x)=x$ and $\lim _{x \rightarrow \infty} \frac{f[x]}{x}=\lim _{x \rightarrow \infty} \frac{h(x)}{x}=1$
  1. A is true, $\mathrm{R}$ is true: $\mathrm{R}$ is correct explanation of $\mathrm{A}$
  2. A, R are true; R is not the correct explanation of $A$
  3. $\mathrm{A}$ is true, $\mathrm{R}$ is false
  4. A is false, R is true

Solution

A: $\lim _{x \rightarrow \infty} \frac{[x]}{x}=\lim _{x \rightarrow \infty} \frac{x-\{x\}}{x}$ $ \begin{array}{ll} =\lim _{x \rightarrow \infty} 1-\frac{\{x\}}{x}=1-0 & \{\because 0 < \{x\} < 1\} \\ = & \lim _{x \rightarrow \infty} \frac{[x]}{x}=1 \end{array} $ $\therefore \quad$ Assertion is true B: $\lim _{x \rightarrow \infty} \frac{f[x]}{x}=\lim _{x \rightarrow \infty} \frac{[x]-1}{x}=\lim _{x \rightarrow \infty} \frac{[x]}{x}=1$ Reason is true

Asked in: AP EAMCET 2023 (18 May Shift 2)

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